Optimal. Leaf size=222 \[ \frac {808 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 (2+3 x)^{7/2}}-\frac {168034 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{5/2}}-\frac {43094 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac {32098184 \sqrt {1-2 x} \sqrt {3+5 x}}{47647845 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac {32098184 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845}-\frac {2036756 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845} \]
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Rubi [A]
time = 0.05, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157,
164, 114, 120} \begin {gather*} -\frac {2036756 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845}-\frac {32098184 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845}+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{9/2}}+\frac {32098184 \sqrt {1-2 x} \sqrt {5 x+3}}{47647845 \sqrt {3 x+2}}-\frac {43094 \sqrt {1-2 x} \sqrt {5 x+3}}{6806835 (3 x+2)^{3/2}}-\frac {168034 \sqrt {1-2 x} \sqrt {5 x+3}}{972405 (3 x+2)^{5/2}}+\frac {808 \sqrt {1-2 x} \sqrt {5 x+3}}{27783 (3 x+2)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx &=\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac {2}{189} \int \frac {\left (-441-\frac {1525 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=\frac {808 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 (2+3 x)^{7/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac {4 \int \frac {-\frac {207611}{4}-\frac {353425 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{27783}\\ &=\frac {808 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 (2+3 x)^{7/2}}-\frac {168034 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{5/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac {8 \int \frac {-\frac {1658793}{8}-\frac {1260255 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{972405}\\ &=\frac {808 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 (2+3 x)^{7/2}}-\frac {168034 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{5/2}}-\frac {43094 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac {16 \int \frac {-1030002-\frac {323205 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{20420505}\\ &=\frac {808 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 (2+3 x)^{7/2}}-\frac {168034 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{5/2}}-\frac {43094 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac {32098184 \sqrt {1-2 x} \sqrt {3+5 x}}{47647845 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {161245065}{16}-\frac {60184095 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{142943535}\\ &=\frac {808 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 (2+3 x)^{7/2}}-\frac {168034 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{5/2}}-\frac {43094 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac {32098184 \sqrt {1-2 x} \sqrt {3+5 x}}{47647845 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}+\frac {11202158 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{47647845}+\frac {32098184 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{47647845}\\ &=\frac {808 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 (2+3 x)^{7/2}}-\frac {168034 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{5/2}}-\frac {43094 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac {32098184 \sqrt {1-2 x} \sqrt {3+5 x}}{47647845 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac {32098184 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845}-\frac {2036756 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845}\\ \end {align*}
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Mathematica [A]
time = 5.03, size = 107, normalized size = 0.48 \begin {gather*} \frac {\frac {24 \sqrt {2-4 x} \sqrt {3+5 x} \left (241253543+1489220097 x+3421407609 x^2+3462531489 x^3+1299976452 x^4\right )}{(2+3 x)^{9/2}}+256785472 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+12066320 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{571774140 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs.
\(2(162)=324\).
time = 0.10, size = 494, normalized size = 2.23
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{137781 \left (\frac {2}{3}+x \right )^{5}}-\frac {168034 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{26254935 \left (\frac {2}{3}+x \right )^{3}}+\frac {1298 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2250423 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {64196368}{9529569} x^{2}-\frac {32098184}{47647845} x +\frac {32098184}{15882615}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {43094 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{61261515 \left (\frac {2}{3}+x \right )^{2}}+\frac {21499342 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{200120949 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {32098184 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{200120949 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(297\) |
default | \(-\frac {2 \left (1361062197 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1299976452 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3629499192 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3466603872 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3629499192 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3466603872 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1613110752 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1540712832 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-38999293560 x^{6}+268851792 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-256785472 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-107775874026 x^{5}-101330034669 x^{4}-23778042336 x^{3}+19087401900 x^{2}+12679220244 x +2171281887\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{142943535 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\) | \(494\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.25, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (1299976452 \, x^{4} + 3462531489 \, x^{3} + 3421407609 \, x^{2} + 1489220097 \, x + 241253543\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{47647845 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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